Optimum design of castellated beams: E ects of composite action and semi-rigid connections

Sharif University of Technology Scientia Iranica
Transactions A: Civil Engineering http://scientiairanica.sharif.edu
Optimum design of castellated beams: Eects of composite action and semi-rigid connections
A. Kaveha; and M.H. Ghafarib
a. Centre of Excellence for Fundamental Studies in Structural Engineering, Iran University of Science and Technology, Narmak, Tehran, P.O. Box 16846-13114, Iran.
b. School of Civil Engineering, Iran University of Science and Technology, Narmak, Tehran, P.O. Box 16846-13114, Iran.
Received 15 May 2016; received in revised form 4 October 2016; accepted 10 December 2016
KEYWORDS Structural optimization; Semi-rigid connections; End lled castellated beams; Composite beams; Colliding Bodies Optimization (CBO); Enhanced Colliding Bodies Optimization (ECBO); Particle Swarm Optimization (PSO).
Abstract. Castellated beams and composite action of beams are widely applicable methods to increase the capacity of the beams. Semi-rigid connections can also redistribute internal moments in order to attain a better distribution. Combination of these methods helps to optimize the cost of the beam. In this study, some meta-heuristic algorithms consisting of the particle swarm optimization, colliding bodies optimization, and enhanced colliding bodies optimization are used for optimization of semi-rigid jointed composite castellated beams. Prole section, cutting depth, cutting angle, holes spacing, number of lled end holes of the castellated beams, and rigidity of connection are considered as the optimization variables. Constraints include the construction, moment, shear, de ection, and vibration limitations. Eects of partial xity and commercial cutting shape of a castellated beam for a practical range of beam spans and loading types are studied through three numerical examples. The eciency of three meta-heuristic algorithms is compared.
© 2018 Sharif University of Technology. All rights reserved.
1. Introduction
Optimizing the structural elements to achieve the best economical and serviceable result is one of the most desirable aims of the structural designers. A typical example of such a trend is the eorts of researchers to eciently increase the capacity of beams. Castel- lated beams and composite beams can increase beams moment of inertia and this eect increases the bending moment capacity of the beams. However, both of them have some limitations and produce secondary eects
*. Corresponding author. Tel.: +98 21 44202710; Fax: +98 21 77240398 E-mail address: [email protected] (A. Kaveh)
doi: 10.24200/sci.2017.4195
that limit their usage. For example, in order to face the secondary shear eect of the castellated beams, some end hole should be lled.
One of the rst studies that evaluated the eect of web opening on composite beams was performed by Redwood and Cho [1]. They also surveyed the failure modes of the hexagonal composite castellated beams. Several tests were performed by Jackson [2] showing that the AISC design guide procedures for composite prismatic beams could be used for calculating the natural frequency of the beams.
The eect of edge constraint component on ex- ural strength of composite beam was studied by Wang and Li [3]. Ellakani and Tablia [4] developed a numerical model for static and free vibration analysis of elastic composite beams with end shear restraint. They found that the end shear restraint played an
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important role when the composite beam interaction was not complete.
There have been several eorts for cost optimiza- tion of composite beam. Morton and Webber [5] optimized the composite I beams. They found that the xed-end conditions led to lighter beams. Senouci and Al-Ansari [6] used the genetic algorithm for opti- mizations.
Optimization of castellated beams with several variables has been performed by many researchers. Sorkhabi et al. [7] optimized the castellated beams by PSO and Genetic Algorithm (GA) and found that PSO was better than GA. Kaveh and Shokohi [8,9] optimized castellated beam with hexagonal and cellular opening by CBO and Charged System Search (CSS) meta-heuristic algorithms. They also considered the eect of end lled plates [10].
Semi-rigid composite connections have been mod- eled and tested by some researchers in recent years. Modeling of beam to girder semi-rigid composite connection with angles, including the eects of con- crete tension stiness, was studied by Oliveira and Batista [11]. They calibrated a modied theoretical model based on the component method and found a good agreement between the experimental and theo- retical results. They observed the role of the concrete before and after crack development and stabilization and the shear lag eect in the slab. Fu et al. [12] mod- eled a 3D nite element model of semi-rigid composite end-plated connections. They presented the eects of longitudinal bar, thickness of the endplate, thickness of beam ange, etc. on capacity of the connection. Gil et al. [13] validated the design procedures experimentally and numerically. They found that the load level in the minor column axis had no in uence on the behavior of the major one. Rex and Easterling [14] developed a simple method of approximating the moment-rotation behavior of composite beam-girder connection. They presented the eect of pre-loading on the moment rotation behavior.
Redistribution of internal member forces by semi- rigid connections can help designers to decrease the total cost of the building. Simoes [15] was one of the rst researchers who optimized frame elements by semi-rigid connections. He used segmental method utilizing linear programming to solve optimization problems. Kameshki and Saka [16] utilized GA to optimize non-linear steel frame with semi-rigid con- nections. They considered P eect in their analysis. Ramires et al. [17] optimized the composite and steel endplate semi-rigid joints by GA. They found that the joint cost could be decreased by 10% by tuning the stiness of the connections. Ali et al. [18] used the multi-stage design optimization by GA for semi-rigid steel frames, and their results showed that the cost of joints might constitute more than
20% of the total cost of an optimized steel frame structure.
The main objective of the present paper is to study the composite action, semi rigidity of joint, and end holes lling on optimization of the castellated beams. The optimization algorithms used in this paper consist of the colliding body optimization, enhanced colliding body optimization, and particle swarm op- timization. Optimization variables are prole section, cutting depth (dh), cutting angle (), holes spacing (s), number of lled end holes of the castellated beam, and rigidity of the connections (Rj).
The present paper is organized as follows: In the next section, design of the castellated beams is intro- duced. In Section 3, design of the composite beams is provided. Semi-rigid connections are introduced brie y in Section 4. Some dependence features of semi rigidity, composite action, and castellated beams are discussed in Section 5. Three optimization algorithms that are used in this study are brie y introduced in Section 6. Based on these sections, problem denition is presented in Section 7. In Section 8, some examples are optimized using dierent metaheuristics, and nally Section 9 concludes the paper.
2. Design of castellated beams
Increasing beam depth without using additional ma- terial by cuttings and welding beams is known as the castellated beam. This method produces a beam web and shape of the holes may be dierent depending on the cutting procedures. Hexagon and circle are two well-known and common shapes of the cutting. To avoid the stress concentration at angles of hexagonal shape, sinusoid cutting can be performed instead of hexagonal openings. Also, for both shapes of holes, secondary plates can be placed between two parts of the beam and increased depth of the beam. In the present study, simple hexagonal shape of castellated beam is considered, because this results in better design in optimization [8-10].
Determining the strength of the castellated beams is more complex than that of the standard beams. Interaction between shear and moment, horizontal shear and radial moment are the main dierences between typical beam design process and castellated beam design procedure.
Unbraced length of the beams plays an important role in their exural capacity. By considering the concrete on the top of the beam, unbraced length can be considered as zero and lateral-torsional buckling can be assumed not to occur.
In the following, the design procedure of the castellated beams is provided based on Load and Resistance Factor Design (LRFD) method of the AISC 360-10 [19].
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Figure 1. Details of a composite castellated beam.
2.1. Flexural capacity Section plastic modulus of a beam identies the mo- ment capacity of the beam and the maximum moment under applied load should satisfy the following condi- tion in AISC 360-10 [19]:
Mu < bMn = bFyZnet - st; (1)
where Mu is the ultimate moment of beam; Mn is the nominal moment capacity of the beam; Znst is the plastic module of steel net section; b is the bending reduction factor; and Fy is the yield strength of steel.
This equation is related to the general beam section. At the holes, Vierendeel mechanism identies the exural demand of the beam. In Vierendeel mechanism, two virtual hinges at top and bottom tee beams between two holes are considered. Vertical shear forces of these two tee beams produce secondary moment that is added to essential moment. Because of linear distribution of moment at depth of the beam, at points \a" and \c" depicted in Figure 1, sum of the two moments may be critical. Therefore, exural capacity of the tee beam should satisfy the following equation in AISC 360-10 [19]:
mu = Vu e
Znet - st + mu
Ztee < bFy; (3)
where mu is the ultimate moment of the secondary shear; Vu is the ultimate shear force; e is the web post length; Mu is the ultimate moment; Znet-st is the plastic module of steel net section; and Ztee is the plastic module of steel tee section. b for concrete and steel is considered to be equal to 0.9.
By lling the hole, Vierendeel mechanism can be neglected.
2.2. Shear capacity Three types of shear forces must be controlled in castellated beam.
First, overall shear capacity of the general section
must satisfy the following equations in AISC 360- 10 [19]:
AW = ds tw; (4)
Vu < vVnw = v 0:6FyAWCv; (5)
where Aw is the area of the net section web; tw is the thickness of the web; ds is height of the internal castellated beam; Vu is the ultimate shear force; Vnw is the nominal web shear capacity of net section; v is the shear reduction factor; and Cv is the web shear coecient.
Second, vertical shear capacity of the tee beams must satisfy the following equations:
Atee = dt tw; (6)
Vu 2 < vVntee = v 0:6FyAteeCv; (7)
where Atee is the area of each tee section and Vntee is the nominal web shear capacity of tee section.
Third, horizontal shear capacity of post web must satisfy the following equations:
Ahe = e tw; (8)
Ig s < vVnp=v 0:6FyAheCv; (9)
where Vh is the horizontal shear at post web; Qg and Ig are the rst and second moments of inertia of general section, respectively; s is the spacing between the holes (Figure 1); Vnp is the nominal shear capacity of post web; Ahe is web post horizontal shear area; and v and Cv are equal to 1.
Because of the penetration of web welding, weld- ing capacity is not critical at horizontal shear check. By lling the hole, horizontal shear can be controlled.
2.3. Web post buckling Due to the horizontal shear, which was discussed above, web post plate performs similar to a cantilevered beam
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without ange, and at one side of it, compression forces produce instability by out of plane buckling [20]. The radial moment and capacity of the web post, according to Structural Stability Research Council (SSRC), must satisfy the following equations:
Cb = 1:75 + 1:05 M1
< bFrb =
375bFy; (10)
where , e, and dh are the cutting angle, hole pure distance, and cutting depth of castellated beam (Fig- ure 1), respectively; tw is the thickness of the web; M1 and M2 are the moment at each beam end; Es is steel module of elasticity; and b is equal to 0.9, similar to the moment equation.
By lling the hole, radial moment can be con- trolled.
3. Design of composite beams
In composite beam with complete interaction, the center line should be found and then moment of inertia of the composite section should be calculated. In this study, temporary shores are considered for use during construction and only composite section is designed for the total live and dead loads. Eective width of the concrete slab should not exceed the limits as described in AISC 360-10 [19]:
1. One-eighth of the beam span, center-to-center of the connections;
2. One-half of the distance to the centerline of the adjacent beam;
3. The distance to the edge of the slab.
According to the height to web thickness ratio, Mn should be determined from the superposition of elastic stresses or from the plastic stress distribution on the composite section. In this study, superposition of elastic stresses is considered because behavior of the composite castellated beam plastic is unpredictable.
In order to consider the eect of dierential shrinkage and creep on a composite steel-concrete structure, eective length can be divided by 3 [21].
Shear studs are designed for transforming shear forces between steel and concrete completely. Shear and strength of the steel channel anchors, without considering compression steel eect, are determined according to AISC 360-10 as follows [19]:
Qu = min(0:85Fcbehc; AsFy) < NcvQn = Nc v 0:3(tsf + 0:5tsw)La
p FcEc; (11)
where Fc and Ec are compression strength and elastic- ity module of concrete, respectively; be and hc are the eective width and height of concrete, respectively; As is the steel section area; tsf , tsw, and La are ange thickness, web thickness, and width of the channel shear studs, respectively. Considering linear shear diagram, Nc is half of the total number of shear studs and v is equal to 0.75.
In positive moment area, compression stress in top of the concrete must be less than allowable compression stress of concrete and tensile stress in the bottom of steel must be less than the allowable tensile stress of steel:
Mu < min(Mncon;Mnst)
= min(b0:7FcZncomtop; bFyZncombot); (12)
where Mncon and Mnst are the nominal moment ca- pacities of composite beam according to concrete part and steel part of the beam, respectively; Zncombot and Zncomtop are section modules of bottom part and top part of the composite net section, respectively.
b for concrete and steel is considered to be equal to 0.9 (AISC 360-10) [19].
In negative moment area, cracked concrete cannot sustain tensile stress and only steel section should be considered as presented in Eq. (1).
The available shear strength of the composite beams will be determined based upon the properties of the steel section alone as presented in Eq. (5), AISC 360-10 [19].
4. Semi-rigid connection
Considering partial xity in connection and redistribu- tion of internal member forces by semi-rigid connec- tion can help designers to decrease the total cost of building, Figure 2. Modeling, analyzing, and testing of this type of connections have been the subject of many research eorts. Linear, bi-linear, and tri-linear moment-rotation models can be adopted to describe the connection behavior. The component method is applied to dene and estimate the properties of the connection [22]. It is usual to consider the web angles subjected to plastic behavior under construction loads
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Figure 2. Model of a semi-rigid connection.
that do not contribute to the initial stiness of the composite joint. Initial stiness of joint (Sj;ini) can be estimated by:
Sj;ini = P Mi
= Fs ds d
Le (dr)
2; (13)
where Mi, Fi, and di are the moment, force, and distance, respectively, and is rotation of semi-rigid joint. Areb, dr, and Le are the area, distance, and eective length of the rebars, respectively.
This equation is based on small rotation of con- nection and calculates the force and moment of the eective component.
In this study, the bilinear model presented in Mur- ray et al. [23] is considered for semi-rigid connection, Figure 3. According to Silva et al. [22], the factor for beam to beam joint is considered 3:
Sj = Sj;ini
; (14)
where Sj is the eective stiness of semi-rigid connec- tion.
After calculating Sj , from the basic equation for calculating rotation of the beam, the xity factor, Rj ,
Figure 3. Bending moment-rotation curve of a joint [23].
for the distributed load is dened as:
Rj = 1
; (15)
where Idef is the eective moment inertia for de ection calculation.
According to this equation, the properties of rebar can be determined. But, in the present study, only optimal xity factor is calculated.
The common shrinkage and temperature rein- forcements parallel to beam can help to have sucient partial xity. If there is no sucient reinforcement, partial xity will be limited.
Moment resistance of the semi-rigid joint must be controlled as:
Mu < bMjRd = bFyrebArebds; (16)
where Mj:Rd is the moment resistance of semi-rigid joint.
5. Semi-rigid composite castellated beam
5.1. De ection of semi-rigid composite castellated beam
Simple equations govern the elastic exural de ection of a typical beam. In the castellated beam, shear defor- mation should be considered. According to Jackson [2], real moment of inertia for de ection is close to the composite moment of inertia of net section. However, AISC design guide for oor vibration indicates that the shear studs are not suciently sti to justify the fully transformed moment of inertia assumption for the composite beams; the following equation determines the eective moment of inertia [23]:
Idef = 0:85In + (Icom 0:85In)
4 ; (17)
where Icom and In are the composite section and net section moments of inertia for de ection calculation. Also, the coecient 0.85 identies the shear eect for typical open web beam.
In negative moment area, three parts are impor- tant for calculating the de ection:
1. De ection from changing end slope of the beam:
def1 = tan Lneg: (18)
2. De ection of semi-cantilevered beam at negative moment part:
def2 = WLneg
3EsIn : (19)
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3. De ection of semi-pinned beam at positive moment part:
def3 = 5WLpos
360 ; (21)
where Lneg and Lpos are the lengths of the negative and positive parts of the beam, respectively.
For concentrated loading, similar calculation must be performed.
5.2. Vibration of semi-rigid composite castellated beam
Frequency of the load carrying system of a oor is the most important factor to identify the level of serviceability of the oor. Frequency is related to the stiness of the beam, boundary condition, and distribution of the mass.
The maximum initial amplitude of the beam is the other important parameter to identify the serviceability level of a oor.
Frequency of beam can be estimated by [24]:
defvib = def1 + def2
+ def31:5 + defcol; (22)
r g
r g
defvib ; (23)
where W and g are the eective weight and gravity acceleration, respectively.
defcol is considered zero and W can be calculated by 0.2 time the live load in addition to dead loads.
The maximum initial amplitude (inch) of the beam is determined as [24]:
Aot =(DLF)max
! ; (24)
+ 2:56 108
: (26)
where Sb is beam spacing and hc is concrete height. (DLF)max values for various natural frequencies are presented in [24].
The required damping ratio for the specied amplitude and frequency is determined by [24]:
Dreq = 35Aof + 2:5…